↑

Properties

Label 2-29-29.17-c8-0-4
Degree $2$
Conductor $29$
Sign $-0.438 - 0.898i$
Analytic cond. $11.8139$
Root an. cond. $3.43714$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.492 − 0.492i)2-s + (72.5 − 72.5i)3-s + 255. i·4-s + 679. i·5-s − 71.4i·6-s − 4.57e3·7-s + (252. + 252. i)8-s − 3.96e3i·9-s + (334. + 334. i)10-s + (−7.36e3 + 7.36e3i)11-s + (1.85e4 + 1.85e4i)12-s − 2.36e3i·13-s + (−2.25e3 + 2.25e3i)14-s + (4.92e4 + 4.92e4i)15-s − 6.51e4·16-s + (2.30e4 − 2.30e4i)17-s + ⋯
L(s)  = 1  + (0.0307 − 0.0307i)2-s + (0.895 − 0.895i)3-s + 0.998i·4-s + 1.08i·5-s − 0.0551i·6-s − 1.90·7-s + (0.0615 + 0.0615i)8-s − 0.604i·9-s + (0.0334 + 0.0334i)10-s + (−0.503 + 0.503i)11-s + (0.894 + 0.894i)12-s − 0.0828i·13-s + (−0.0586 + 0.0586i)14-s + (0.973 + 0.973i)15-s − 0.994·16-s + (0.275 − 0.275i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.438 - 0.898i$
Analytic conductor: \(11.8139\)
Root analytic conductor: \(3.43714\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :4),\ -0.438 - 0.898i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.691808 + 1.10778i\)
\(L(\frac12)\) \(\approx\) \(0.691808 + 1.10778i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-4.24e5 - 5.65e5i)T \)
good2 \( 1 + (-0.492 + 0.492i)T - 256iT^{2} \)
3 \( 1 + (-72.5 + 72.5i)T - 6.56e3iT^{2} \)
5 \( 1 - 679. iT - 3.90e5T^{2} \)
7 \( 1 + 4.57e3T + 5.76e6T^{2} \)
11 \( 1 + (7.36e3 - 7.36e3i)T - 2.14e8iT^{2} \)
13 \( 1 + 2.36e3iT - 8.15e8T^{2} \)
17 \( 1 + (-2.30e4 + 2.30e4i)T - 6.97e9iT^{2} \)
19 \( 1 + (1.14e5 - 1.14e5i)T - 1.69e10iT^{2} \)
23 \( 1 - 4.56e5T + 7.83e10T^{2} \)
31 \( 1 + (6.67e4 - 6.67e4i)T - 8.52e11iT^{2} \)
37 \( 1 + (1.24e6 + 1.24e6i)T + 3.51e12iT^{2} \)
41 \( 1 + (-889. - 889. i)T + 7.98e12iT^{2} \)
43 \( 1 + (-1.32e6 + 1.32e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (3.55e5 + 3.55e5i)T + 2.38e13iT^{2} \)
53 \( 1 + 5.74e6T + 6.22e13T^{2} \)
59 \( 1 + 7.85e6T + 1.46e14T^{2} \)
61 \( 1 + (1.53e7 - 1.53e7i)T - 1.91e14iT^{2} \)
67 \( 1 + 2.10e7iT - 4.06e14T^{2} \)
71 \( 1 - 4.75e7iT - 6.45e14T^{2} \)
73 \( 1 + (-2.33e7 - 2.33e7i)T + 8.06e14iT^{2} \)
79 \( 1 + (1.91e7 - 1.91e7i)T - 1.51e15iT^{2} \)
83 \( 1 + 6.18e7T + 2.25e15T^{2} \)
89 \( 1 + (-6.13e7 + 6.13e7i)T - 3.93e15iT^{2} \)
97 \( 1 + (-5.90e7 - 5.90e7i)T + 7.83e15iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.63240889468163290425111935130, −14.22219877867577989743092739575, −12.99202139337818589962246817259, −12.54356822486946518760343772793, −10.48638791640861280640537833717, −8.920243071214547034587114287998, −7.38502034870868210028991427322, −6.70770776555130876423193617873, −3.32769022370308738097542145780, −2.62725347082973396181183105470, 0.48828706133913205190763403665, 3.02784666037565489199838489773, 4.71947479887581635099953905339, 6.35187099071187072177750302009, 8.839414038953676353667577653074, 9.468481766417758206801020946148, 10.54623759910324629156685834685, 12.77406423919493867468289548996, 13.66039236762371660058881017455, 15.21846724073966562397167482569

Graph of the $Z$-function along the critical line